User talk:Vel!/BEAF
There's a lot of talk going around of BEAF beyond \(\varepsilon_0\) being ill-defined. Has anyone yet found an error in this definition? you're.so. 15:41, June 2, 2014 (UTC) All I can give you is my opinion based on my own personal work on BEAF. There appears to be a non-ambiguous interpretation of BEAF up to e0, and I even formalized it at one point in my personal writings. This was achieved some time around 2007-2008 before the inception of my site. Much effort was put out since then to figure out how to go further. The result has been no end of confusion. I believe that in principle it is possible to create a system for legion space. The reason is because eventually ''a block of entries has to reach any desired size. If you keep multiplying by the ''prime ''eventually you can get any number of entries in a block. However there are technical difficulties to formalizing this concept. Some of these technical difficulties can in principle be overcome, though the solutions may not be particularly elegant. However it is not entirely clear that there may not be some "hitch" down the road. Bowers' simply assumed that the array structures existed, that they made sense, and that there was a unique non-ambiguous way to interpret them. In short, there is no way to know whether or not it's ill-defined until someone finds definite proof that something is definitely unworkable or someone demonstrates that an explicit definition of BEAF can be formulated. We could ''debate ''whether or not it's ill-defined, but let me suggest a better way to settle the debate. If it's well-defined, then someone needs to create an explicit definition of it to demonstrate this. If it's ill-defined then someone needs to find a definite reason why Bowers' idea is ambiguous or problematic. Short of this we are merely speculating. Sbiis Saibian (talk) 16:31, June 2, 2014 (UTC) :I would say that there generally isn't a "definite reason" why what someone says is ambiguous, other than the fact that you read/hear what they write/say, and you can tell that they are missing details so that you can't pinpoint specifically what they mean. After reading Bowers' description of legion arrays, it is clear to me that he hasn't precisely defined the notation - no definite reason beyond that. :Of course, someone could define a notation that seems like it could be what Bowers really meant by array spaces and legion spaces. But, I'm willing to bet that there are multiple ways to due this, due to the very ambiguity that we are trying to eliminate. So I don't think one can come up with a notation and say, "''This is how fast Bowers' notation grows". Deedlit11 (talk) 07:42, June 5, 2014 (UTC) Definition "An array is a function A : W -> W\{0} where the number of outputs greater than one is finite." What does that mean, actually? And what does it output? Wait nvm. So it can handle limit ordinals now? King2218 (talk) 23:44, June 2, 2014 (UTC) :Old news, man. Get with the times. you're.so. 01:30, June 3, 2014 (UTC) :What I'm saying is that you can probably figure it out from the page. you're.so. 01:50, June 3, 2014 (UTC) :Nvm, I already figured that out before you posted :P King2218 (talk) 02:06, June 3, 2014 (UTC) Stewardess Problem It must be at the end. (try doing some analysis to see for yourself) King2218 (talk) 18:19, June 9, 2014 (UTC) :Can you be more clear? you're.so. 01:42, June 10, 2014 (UTC) :Okay. See how \{\omega,\omega,\omega\} in your definition is the limit of \{\{\omega,1,\omega\},\{\omega,2,\omega\},\{\omega,3,\omega\},...\} . \{\omega,1,\omega\} is equal to \omega so no problem there. \{\omega,2,\omega\} is the limit of \{\{\omega,2,1\},\{\omega,2,2\},\{\omega,2,3\},...\} , \{\omega\uparrow2, \omega\uparrow\uparrow2,\omega\uparrow\uparrow\uparrow2,...\} , or basically \{\omega\cdot\omega,\omega\uparrow\omega,\omega\uparrow\uparrow\omega,...\} . The limit of those is \omega\uparrow^\omega\omega or \omega\{\omega\}\omega which makes it inconsistent with Bowers's definition. King2218 (talk) 11:53, June 10, 2014 (UTC) ::Gotcha. I've changed the definition accordingly, thanks for the correction. you're.so. 15:26, June 10, 2014 (UTC) Larger structures While dealing with normal arrays like \{\omega,\omega,1,2 (1) \omega (2) 5\} might not be problem, you didn't define the rules for something like \{\omega,5,2\} & \{\omega,1,1,2\} & \omega . I don't believe that the power of sub-legion BEAF is \psi(\Omega_\omega) . The variant of BEAF, considered by Hyp cos, uses arrays with more than \omega entries (if we let X = \omega ) and yet he used structures X_n , which is the core of the power and nether Bowers nor you mention it. Also, why the article doesn't explain rules for past-legion spaces at all? How \{\omega,\omega+1 / 2\} & n must be solved for instance? I know that it must be somehow translated into your ternary & operator, but it isn't included into definition. Ikosarakt1 (talk ^ ) 07:27, June 10, 2014 (UTC) :Because the ternary & operator is merely a notational shorthand for something that's already in the definition of this function. I'm in a hurry now, so I'll probably answer more elaborately later you're.so. 15:42, June 10, 2014 (UTC) :So / symbol isn't in the notation, so your definition doesn't work past sub-legions and incomplete. I propose to remove all conjectures past \{\omega,\omega / 2\} for now. The fact that \{\omega,\omega / 2\} is the fixed point of n \mapsto &(a,b,n) says nothing about, say, lugiattic arrays or even legiattic. There would be further extensions of & operator like &&, &&&, @, %, # (Bowers mentioned them all). Probably you must define ternary analogues of them. Ikosarakt1 (talk ^ ) 06:10, June 12, 2014 (UTC) ::We can have a 5-argument function &(a,b,c,d,e), d is the number of symbols, and e the type of the symbol. Wythagoras (talk) 06:43, June 12, 2014 (UTC) ::(ec) Nnnnnnnnnot quite. You're concluding too fast here. :::"So / symbol isn't in the notation" ::It isn't, because (this is important) the function defined on the page isn't actually a notation. It's a function that operates on arrays, and we can use a notation to represent those arrays. ::Problems arise when a "function" exists only in the form of a poorly-defined notation. And that's what BEAF is beyond \(\varepsilon_0\). :::"so your definition doesn't work past sub-legions and incomplete" ::The function does "work"...it's just that the process of finding the mapping between our ordinals and what Bowers could have meant gets seriously murky at that point. I can pull any old large ordinal out of my back pocket and plug it into the function, and it'll return a value, but the real question is where that ordinal fits in into Bowers' intuition, if at all. :::"The fact that \{\omega,\omega / 2\} is the fixed point of n \mapsto &(a,b,n) says nothing about, say, lugiattic arrays or even legiattic. There would be further extensions of & operator like &&, &&&, @, %, # (Bowers mentioned them all)." ::You seem to have a preconceived notion that we need to define and explicitly name all these operators to be able to extend our function to these types of arrays. Turns out, we don't. (We don't even need to define the & function — I just put that out there as a proof of concept.) All we need to do is to specify ordinals and make convincing arguments that they line up with Bowers' speculations. you're.so. 06:56, June 12, 2014 (UTC) :Then I guess that BEAF works like FGH (we have a sketch of definition for all countable ordinals, but fundamental sequences are need to be defined.) Can BEAF reach uncomputable ordinals? Ikosarakt1 (talk ^ ) 07:12, June 12, 2014 (UTC) ::Yes and no. In one sense, we can insert whatever ordinals we like and we will get a properly defined result. \(\{3,3(\omega_1^\text{CK})2\}\) is valid, as long as we have an FS for the Church-Kleene ordinal. (Although it's not actually that large, since the Kleene's O fundseq gets a pretty slow start. \(\{10,10^{100}(\omega_1^\text{CK})2\}\), I expect, would be more comparable to outputs of the busy beaver function.) Deedlit developed a version of BEAF that allows this for finite inputs and ordinal structures (and I helped to simplify it). Sbiis, however, has argued that our kinds of results only emulate the rules of BEAF without explaining the structures, which is where all the money's at. ::But in another sense, we've created a rule that forbids us from jumping ahead like that. This is the condition of "constructibility" as explained on the page. Constructibility creates a self-contained system where we can only reuse ordinals that we already have. If we restrict ourselves to constructible arrays, we can't jump ahead to \(\omega_1^\text{CK}\). (I am 100& certain that there is no way to construct that ordinal!) So why would we want to add this seemingly unnecessary rule if it restricts the power of the function? It's because we are trying to emulate the "bottom-up" method of construction that Bowers uses. Bowers didn't get to jump ahead to \(\varepsilon_0\) immediately; instead he showed us how linear arrays work, then multidimensional ones, then superdimensional ones, and then he showed how \(\varepsilon_0\) comes naturally as a diagonalization over all those concepts. ::Perhaps surprisingly, limiting ourselves with constructibility tells us a lot more about BEAF. It is my belief that the supremum of all BEAF-constructible ordinals is the same as the ordinal for L-space. But wait, doesn't that mean that the function, when restricted by constructibility, is less powerful than BEAF? Yes. Yes it does, but I believe we can easily extend the concept of constructibility to make it far more powerful — probably further than Bowers ever went, and without losing formality. However, I'd really prefer that we figure out the smaller ordinals first :3 ::As a sidenote, it looks like someone (you, maybe?) changed the definition of constructibility on the page. I've reverted it because it's wrong. We don't need the array-of operator, which is again a mere notational shorthand. you're.so. 16:01, June 12, 2014 (UTC) Analysis between \(\{\omega,\omega (1) 2\}\) and \(\{\omega,\omega,2 (1) 2\}\) Let \{\omega,\omega (1) 2\} = \psi(\Omega^{\Omega^\omega}) Then, if indeed \{\omega,\omega+n (1) 2\} = \{\{\omega,\omega+n-1 (1) 2\},\{\omega,\omega+n-1 (1) 2\},\cdots,\{\omega,\omega+n-1 (1) 2\}\} (with n+1 \{\omega,\omega+n-1 (1) 2\} 's): Then \{\omega,\omega*2 (1) 2\} would be equivalent to \psi(\Omega^{\Omega^\omega}*2) and nowhere close to \psi(\Omega^{\Omega^{\omega*2}}) . Hyp cos proposed to represent intermediate ordinals using entries at transfinite positions, but by our definition that's disallowed. By that we can come to \{\omega,\omega,2 (1) 2\} is merely \psi(\Omega^{\Omega^\omega+1}) and nowhere LVO. Ikosarakt1 (talk ^ ) 07:47, June 10, 2014 (UTC) :Why \{\omega,\omega+n (1) 2\} = \{\{\omega,\omega+n-1 (1) 2\},\{\omega,\omega+n-1 (1) 2\},\cdots,\{\omega,\omega+n-1 (1) 2\}\} ? How does that follow form the definition? Wythagoras (talk) 14:50, June 10, 2014 (UTC) :\(\{\omega,\omega+n (1) 2\} = \{\{\omega, \omega (1) 2\}, n + 1 (1) 2\}\) by the Infinite Catastrophic Rule, not \(\{\{\omega, \omega + n - 1 (1) 2\}, n + 1 (1) 2\}\). you're.so. 15:30, June 10, 2014 (UTC) :I think \{\omega,\omega,2(1)2\} is \{\theta(\Omega^{\omega}+1)\} . I dunno. (Also, Ikosarakt I did something wrong and I got that \{\omega,\omega,4\} = \zeta_1 which should be/is \varphi(3,0) ) King2218 (talk) 14:06, June 11, 2014 (UTC) :Certainly that's correct. If we want to prove that the limit of sublegion BEAF is LVO, we have to prove two strong claims: 1: If \{\omega,\omega \#\} = \theta(\alpha,0) Then \{\omega,\omega*(1+\beta) \#\} = \theta(\alpha,\beta) 2: If the ordinal is in the form \{\omega,\omega,1,1,\cdots,2\} , where "2" is at 3+\alpha -th position, then it is equal to \theta(\Omega^\alpha) . Ikosarakt1 (talk ^ ) 14:52, June 11, 2014 (UTC) :Explain what "#" means. you're.so. 16:44, June 12, 2014 (UTC) :You're joking? We all know that # is just the remainder of the array. Ikosarakt1 (talk ^ ) 19:30, July 25, 2014 (UTC) MathJax Can you allow to replace MathJax to LaTeX? Ikosarakt1 (talk ^ ) 07:47, June 10, 2014 (UT Copilot I believe the current definiton is wrong. Saying κ=π-1 is just like saying p=1. Wythagoras (talk) 14:50, June 10, 2014 (UTC) I don't see what's wrong with this. π is the position of pilot, and copilot is located right before pilot (if it exists), thus making the position of the copilot κ=π-1. I don't see where p is involved here. LittlePeng9 (talk) 15:02, June 10, 2014 (UTC) :Yes, the position of the copilot is π-1, but the value is A(π-1). Therefore I say that saying κ=π-1 is just like saying p=1. (The position of p is 1, but the value is A(1).) Wythagoras (talk) 15:06, June 10, 2014 (UTC) ::But we define p to be value of prime entry, and κ is position of copilot. LittlePeng9 (talk) 15:10, June 10, 2014 (UTC) b and p are the values of the base and prime, while π and κ are the positions of the pilot and copilot. (It's also why I used Latin letters to denote the first two and Greek ones to denote the latter two.) you're.so. 15:27, June 10, 2014 (UTC) \{\omega,\omega+1,\omega\} Look what happens: \{\omega,\omega+1,\omega\} = \{\{\omega,\omega,\omega\},2,\omega\} Now it's impossible to resolve it by any rule in the ruleset because prime is finite and pilot is infinite (we can't decrement it.) So the ruleset is incomplete. Ikosarakt1 (talk ^ ) 19:30, July 25, 2014 (UTC) :Limit Rule applies. you're.so. 21:22, July 25, 2014 (UTC) :Thanks. So \{\{\omega,\omega,\omega\},2,\omega\}n = \{\{\omega,\omega,\omega\},2,n\} = \{\{\omega,\omega,\omega\},n,n-1\} Ikosarakt1 (talk ^ ) 05:32, July 26, 2014 (UTC) Wait, what is \{\omega,\omega+1,\omega\} ? King2218 (talk) 08:25, August 3, 2014 (UTC) \{\omega,\omega+1,\omega\} = \phi(\omega,1) , and surprisingly \{\omega,\omega+1,\omega\} = \{\omega,\omega+m,\omega\} = \{\omega,\omega*2,\omega\} . So it's still consistent with \{\omega,\alpha*(1+k) \#\} = \theta(\beta,\alpha) . Ikosarakt1 (talk ^ ) 09:04, August 3, 2014 (UTC) Array-of function redefinition What if we didn't care about the ordinal equivalents of BEAF structures? For example, we wanted to compute for the value of triakulus, which is \(3 \text{&} 3 \text{&} 3\) or \(\text{&}(3, 3, \text{&}(3, 3, 3))\). The problem is, \(\text{&}(3, 3, 3)\) returns an array, and \(\text{&}\) takes only countable ordinals, and not arrays, as arguments. My new definition could fix this problem. (I'm not editing the main page because I'm not entirely sure if this works). \(O(\gamma)\) is defined as follows: *If \(\gamma\in\mathbb N\), \(O(\gamma) = O(\{0\mapsto\gamma\})\) *\(\alpha = \{\beta|\gamma(\beta)\in\mathbb N\backslash\{0, 1\}\}\) *Define \(\gamma'\) as \(\gamma\) but, for all \(n\in\alpha\), \(\gamma'(n) = \omega\) *\(O(\gamma) = \gamma'\) \( \text{&}(\alpha, \beta, \gamma) = \begin{cases} \text{&}(\alpha, \beta, v(O(\gamma))), & \gamma\text{ is an array} \\ \{0\mapsto\alpha, 1\mapsto\beta, \gamma\mapsto2\}, & \text{otherwise} \end{cases} \) Looks good? King2218 (talk) 05:16, July 29, 2014 (UTC) (poll removed) :But an array evaluates to an ordinal, so this isn't necessary. However, the mistake that you pointed out by the poll above can be fixed as following: \(\text{&}(\alpha,\beta,\gamma) = \begin{cases} \{0\mapsto\alpha, 1\mapsto\beta, \gamma\mapsto2\}, & \text{if } \gamma \in \text{Lim} \\ \{0\mapsto\alpha, 1\mapsto2, \gamma\mapsto2\}, & \text{otherwise} \end{cases} \) Wythagoras (talk) 18:47, July 29, 2014 (UTC) :Arrays don't evaluate to ordinals, the v() function does it for them. King2218 (talk) 10:32, July 30, 2014 (UTC) All that I can say is that if you think \(\&(3,3,3) = \{3,3,3\}\) should be true, you don't understand what the function is supposed to do. The third argument of the function specifies a type of structure being created, and the first two present a base and prime respectively. In usual BEAF jargon, we don't think of "3 structures," rendering \(\&(3,3,3)\) a bit odd to use against the backdrop of Bowers' terminology. To properly emulate the two-argument operator, the third argument should be a limit ordinal. When we say \(3 \& 3\), under the hood we're saying "an X structure with base 3 and prime 3," or \(\&(3, 3, \omega) = \{3, 3 (1) 2\} = \{3, 3, 3\}\). you're.so. 06:15, July 30, 2014 (UTC) :I guess I should change it then. Thanks for the clarification. King2218 (talk) 10:32, July 30, 2014 (UTC) :What about \(\&(3,3,\omega+1)\) X+1&3 is equal to latri, so it should be {3,3(1)3}. However FBs definition gives {3,3(1)1,2} = {3,3(1)latri} Wythagoras (talk) 10:31, July 30, 2014 (UTC) :Whose definition? King2218 (talk) 10:32, July 30, 2014 (UTC) :FBs. Wythagoras (talk) 10:45, July 30, 2014 (UTC) ::@wyth Why should it be equal to {3,3(1)3}? By the way, you did the Catastrophic Rule completely wrong -- it's {3,3(1)1,2} = {3,3,3(1){3,2(1)1,2}} = {3,3,3(1){3,3,3(1)3}}. you're.so. 12:30, July 31, 2014 (UTC)